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Mirrors > Home > ILE Home > Th. List > sb7af | GIF version |
Description: An alternate definition of proper substitution df-sb 1686. Similar to dfsb7a 1911 but does not require that 𝜑 and 𝑧 be distinct. Similar to sb7f 1909 in that it involves a dummy variable 𝑧, but expressed in terms of ∀ rather than ∃. (Contributed by Jim Kingdon, 5-Feb-2018.) |
Ref | Expression |
---|---|
sb7af.1 | ⊢ Ⅎ𝑧𝜑 |
Ref | Expression |
---|---|
sb7af | ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sb6 1807 | . . 3 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑧 → 𝜑)) | |
2 | 1 | sbbii 1688 | . 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧]∀𝑥(𝑥 = 𝑧 → 𝜑)) |
3 | sb7af.1 | . . 3 ⊢ Ⅎ𝑧𝜑 | |
4 | 3 | sbco2 1880 | . 2 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑) |
5 | sb6 1807 | . 2 ⊢ ([𝑦 / 𝑧]∀𝑥(𝑥 = 𝑧 → 𝜑) ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) | |
6 | 2, 4, 5 | 3bitr3i 208 | 1 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑧(𝑧 = 𝑦 → ∀𝑥(𝑥 = 𝑧 → 𝜑))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 ∀wal 1282 Ⅎwnf 1389 [wsb 1685 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 |
This theorem is referenced by: dfsb7a 1911 |
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